They’re not perfect synonyms, but they are similar.
“Elaborate” is something that has a lot of details and is very intricate. It COULD be something that is complicated. However, it often is not. For example: “Wow! Look at that elaborate costume!” Example: a wedding cake, a chandelier, the details on how to care for your friends dog.
“Complicated” means there are many parts, and it’s usually also confusing. Examples: “Finding my way through the complicated subway system caused me to be late!” “This complicated math problem has taken me 45 minutes!” “This recipe shouldn’t be so complicated!”
Also, please note that “elaborate” is an adjective and also a verb. It is pronounced and used differently in each part of speech.
Please let me know if you have any other questions.
Happy New Year!!